mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a paraproduct is a non-commutative bilinear operator acting on functions that in some sense is like the product of the two functions it acts on. According to

Svante Janson Carl Svante Janson (born 21 May 1955) is a Swedish mathematician. A member of the Royal Swedish Academy of Sciences since 1994, Janson has been the chaired professor of mathematics at Uppsala University since 1987. In mathematical analysis, Jans ...

and Jaak Peetre, in an article from 1988, "the name 'paraproduct' denotes an idea rather than a unique definition; several versions exist and can be used for the same purposes." The concept emerged in J.-M. Bony’s theory of paradifferential operators. This said, for a given operator

\Lambda

to be defined as a paraproduct, it is normally required to satisfy the following properties: * It should "reconstruct the product" in the sense that for any pair of functions

(f, g)

in its domain, ::

fg = \Lambda(f, g) + \Lambda(g, f).

* For any appropriate functions

f

and

h

with

h(0)=0

, it is the case that

h(f) = \Lambda(f, h'(f))

. * It should satisfy some form of the

Leibniz rule Leibniz's rule (named after Gottfried Wilhelm Leibniz) may refer to one of the following: * Product rule in differential calculus * General Leibniz rule, a generalization of the product rule * Leibniz integral rule * The alternating series test, al ...

. A paraproduct may also be required to satisfy some form of

Hölder's inequality In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of spaces. :Theorem (Hölder's inequality). Let be a measure space and let with . ...

Notes

Further references

* Árpád Bényi, Diego Maldonado, and Virginia Naibo
"What is a Paraproduct?"
''Notices of the American Mathematical Society'', Vol. 57, No. 7 (Aug., 2010), pp. 858–860. Bilinear maps {{algebra-stub